Convergence of the Variable Order and Variable Stepsize Direct Integration Methods for the Solution of the Higher Order Ordinary Differential Equations

The DI methods for directly solving a system of a general higher order ODEs are discussed. The convergence of the constant stepsize and constant order formulation of the DI methods is proven first before the convergence for the variable order and stepsize case. 1. INTRODUCTION Many problems in engineering and science can be formulated in terms of such a system. The general system of higher order initial Some of the higher order ODEs problems found values ODEs is given by in'the literature are on the symmetrical bending of a laterally loaded circular plate, the bending Y ^ i' = f / Y) ' = 1 ? s (1) of a thin beam clamped at both ends. Both nrob-' ' ' lems are given in Russell and Shampine (1972). with initial conditions Others are on the steady flow of a viscoelastic Y(a \ = v fluid parallel to an infinite plane surface with uniform sunction; given in Serth (1975), control in the interval a < x < b, where the i-th equa-theory in Enright et al (1974) cop i anar c i rcu lar tionisoforderdjand two body motion in shampine and Gordon /ft *\ ^ _ j\ (1975), gravitational n-body problem in Dol et